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458 Sir W. D. Niven. [Mar. 22, 

first time that the kite has escaped undamaged and the trace been decipher- 
able. In general a steady and uniform wind is associated with steady 
temperature conditions, but when the temperature at a given height is 
subject to much fluctuation, so that the meteorograph registers different 
temperatures each time it passes through that height, the wind also is usually 
variable in direction and velocity, but these latter conditions are not 
necessarily accompanied by a steep temperature gradient. 



The Calculation of Ellipsoidal Harmonics. 
By Sir W. D. Niven, K.C.B, V.-P.RS. 

(Eeceived March 22,— Eead March 29, 1906.) 

1. The object of this note is to show how ellipsoidal harmonics of the fourth, 
fifth, sixth, and seventh degrees may be calculated. Some of the fourth and 
fifth degrees are easily found, depending as they do upon the solution of a 
quadratic equation. When, however, the type of the harmonic of the fourth 
degree is ©i ©2 where, if we employ Greek letters for current co-ordinates, 

£2 v 2 £2 

(H), = k u-Jl u__J> 1 

1 a 2 + 6i b 2 + 6 1 c 2 + 0! ' 



71 2 t 2 

2 a 2 + 2 b 2 + 2 e 2 + 0% ' 

and, in order that ©1 ©2 may satisfy Laplace's equation, the quantities 0i, 02 
are to be found from 

1114 

1 j — + i__ + __Z = (i) 



a 2 + 1 J ) 2j t i c 2j r i ei _0 2 

1114 



a 2 + 2 b 2 + 2 c 2 +0 2 2 -0i 

the elimination of 0% leads to a sextic equation in 0\. Since, however, the 
roots occur in pairs as in (1) and (2), if we put 

0i + 02 = 2%, 1 -0 2 = 2v > (3) 

the equations for u and v derived from (1) and (2) will be of lower degrees 
than the sixth. 

When the substitutions (3) are made in (1) the latter becomes 

1 + * , + 3 X , +-=0, (4) 



a 2 ~\-u + v b 2 -\-u-\-v e 2 + u-\-v v 
and equation (2) will be the same with the sign of v changed 



1906.] The Calculation of Ellipsoidal Harmonics. 459 

Multiplying up in (4), arranging in powers of v and putting Pi for the sum 
of the quantities a 2 + % b 2 + u, c 2 + u, P 2 for their sum taken two and two and 
P 3 for their product, we find 

5v s + 4P^ 2 + 3P 2 v + 2P 3 = 0. (5) 

As this equation is true when — v is entered for v, it follows that 

^=-|P 2 =-ig. (6) 

Hence 5P 3 = 6P1P2. (7) 

This is a cubic in u which when solved leads to the values of v correspond- 
ing to those of u. 

2. To express the cubic in a convenient form let a, b, c be in ascending 
order of magnitude and write 

b 2 —a 2 , a 2 + ti, v = (p, x, y) (c 2 — a 2 ). (8) 

Then will 

b 2 + n, c 2 + u = (x+p, x+1) (c 2 —a 2 ), (9) 

and Pi = (3® + 1 +p) (c 2 - a 2 ), 

P 2 = [3x 2 + 2(l+p)x+p](c 2 -a 2 ) 2 , > (10) 

P 3 = \x z + (1 +p) x 2 +px] (c 2 — a 2 ) 3 . ^ 

Entering these values in (7), we obtain 

4:9x^ + 4:9 (l+p)x 2 + (12 + 37p + 12p 2 )x+6p(l+p) = 0. (11) 

As the roots of this equation are all real the equation may be solved * by 
the method given in treatises on trigonometry. 

By putting 

« =-1(1 +*) + *, (12) 

we obtain X 3 — $X — r = 0, (13) 

where g = -^V (1 -p +p 2 \ r = -p^ (1 +p) (2 - hp + 2p 2 ). 

The solution is then 

X = [COS a, COS (f 7T + a), COS (§-7T — a)] a/ -J£ } 

/ 27 

where cos 3a = 4r a / - — . . 

V 64g 3 

The corresponding expression for the difference of the roots is given by 

y* = -%\Sa?+2(l+p)x+p'\ = $(l-p+p?-9X*). (14) 

The equations (12), (13), (14) completely determine the values of and they 
show that there are three harmonics of the type considered. 

3. Harmonics of the types (f, y, £ rj^ (& &> &£) ©i ®2 may be calculated 



460 Sir W. D. Niven. [Mar. 22, 

in similar fashion, but the working being in all respects like that in §§ 1, 2 
need not be repeated. Equations (1) and (2) will, of course, be different. 
For instance, if we are considering x ©x © 2 the first term in equations (1) and 
(2) must be multiplied by 3, and if we are considering yz ©i© 2 the second and 
third terms must be multiplied by 3. 

The results for the seven harmonics described above will now be stated in 
the final forms suitable to the trigonometrical solution of §2, i.e., in the forms 
similar to those expressed by the three equations (12), (13), (14). 

£©1©2. 

27*= -10(l+j)) + 27X; 

(27X) 3 -3 (28-43p + 28p 2 ) (27X)-(1 +p) (160-355jo4 160p 2 ) = ; 

11{21yf = 27 2 (16-23^ + 16p 2 )-[ll(27X)-2(l+i>)] 2 . 

1?©1©2. 

27*= -(10 + 7i?)+27X; 

(27X) 3 -3(28-13p+13j? 2 )(27X)+(2p-l)(160 + 35p-35p 2 ); 

77(27y) 2 = 27 2 (16-9j? + 9p 2 )-[ll(27X)+4p-2] 2 . 

£©1© 2 .— 

27*= -(7 + l(hV)+27X; 

(27X) 3 -3(13-13^ + 28j? 2 )(27X)-(2-,p)(35-35j?-160p 2 ); 

77(27</) 2 = 27 2 (9-9p + 16p 2 )-[ll (27X)+4-2^] 2 . 

?;t©i©2. — 

33*= -10(l+p) + 33X; 

(33X) 3 -21 (4-^+4^ 2 )(33X)-(l+i>)(160-463^ + 160p 2 )= 0; 
117 (33y) 2 = 33 2 (16-7i?+16^ 2 )-[13(33X)+2(l+p)] 2 

33*= -(10 + 13p)+33X; 

(33X) 3 -21 (4+7i?-7j9 2 )(33X)-(l-2 i3 )(160 + 143p-143jo 2 ) = 0; 

117 (33y) 2 = 33 2 (16-25^+25p 2 )-[13 (33X)+2-4p] 2 . 

f?7@i©2. — 

33*= -(13 + 10p) + 33X; 

(33X) 3 -21(7-7i? + 4p 2 )(33X)-(2-p) (143-143^-160/) = 0; 

117 (33y) 2 = 33 2 (25-25p + 16p 2 )-[13 (33X)-4+2^] 2 . 

39*= -13(l+.p)+39X; 
(39X) 3 - 147 (1 -p +p 2 ) (39X)- 143 (1 +p) (2-5p + 2p 2 ) = ; 

33y 2 = 5(1— p+_p 2 -9X 2 ). 



1906.] The Calculation of Ellipsoidal Harmonics. 461 

4. The results given in the preceding section exhaust all the cases in 
which the harmonic has two <& factors. We pass on to the harmonic of 
sixth degree with three such factors 

© 1 ©2©3. 

We have to solve the following set of equations : — 

1114 4 



d 2 +0 1 &+$! G 2 +0 X 0!-0 2 0!-0 3 

1 1 1 4 4 /-.^x 

+ 15 7r + ~9 J — /T + d "n"^"a B~ = 0* (16) 



^ 2 +0 2 6 2 +0 2 ^+02 02-01 02-03 

1 +x9-^+-T^+7r^ + ^-^-=°- (17) 



^ +03 5 2 +#3 C 2 +0 3 03-01 03- # 3 

Following the method of § 1, we now put 

01 + 02+03 = 3%, -\ 

2 _0 3 = 3r, 
8 -0i = 3s, 
ft-02 = 3*. J 
And, with these substitutions, equation (15) becomes 



y (is) 



a 2 + ^ + ^— 5 & 2 + ^ + £ — s c 2 + u + t — s 2>\t 
Or, on multiplying up, 

4(^-s)P3+[4(^-s) 2 ^3^]P2 + 2(^-s)[2(^-s) 2 -3^]Pi 

+ (*-*)* [4 (j_ g )8_9k] - o. (20) 
Similarly, 

4 (*•-*) P 3 + [4(r-^-3^] P 2 +2 (r-*) [2 (V-*) 2 - 3 ^] Pi 

+ ^-^[4^-^-9^] = 0. (21) 

The third equation need not be written. 

In these equations Pi, P 2 , P3 have the same meanings as in § 1, except that 
w is now the mean of three 0's instead of two. As the determination of 
r, s, t from the equations appears not to be practicable, Pi, P 2 , P3 will be 
found in terms of r, s, t. . 

First eliminate P 3 from (20) and (21) observing that r+s + t= 0. There 
results 

(3^ + 2r8)P a + 2(s-r)(r-0^-*)Pi = 6 (r~t)(t-s) (t 2 -rs). (22) 
It will be noticed that t 2 — rs is a symmetrical function of r, s, t, for since 
r + s + ^=;0itis| (r 2 + s 2 + ^ 2 ). This quantity, as it appears frequently in the 
work, will be denoted by V. 



462 Sir W. D. Niven. [Mar. 22, 

CombiniDg (22) with another equation of the same form, with the letters 

interchanged, we find 

P 2 = -J/V, (23) 

(s-r) (r-t) (t-s) Pi = - IY 2 . (24) 

To obtain P 3 subtract (21) from (20), still making use of r + s + t = 0. There 

results 

4P 3 -5 ( 5 _ r ) p 2+ i6VPi-0-r) (21* 2 -fllV) = 0. (25) 

Writing two other similar equations and adding the three we find, on 
removing the factor 3, 

4P 3 + 16VPi + 7(s-r)(r-0(*- 5 ) = 0. (26) 

The elimination from (23), (24), (26) of V and the product of the differences 
of r, s, t may now be made and we obtain 

48Pi (9P 3 -10P 1 P 2 )-35P 2 2 = 0. (27) 

This is a biquadratic in u, showing that there are four harmonics of the type 

under discussion. 

The equation giving r, s, t must now be found. It must clearly be'of the 

form 

X 3 + T 5 8-P 2 X + E = 0, (28) 

where r + s + t = 0, 

st + tr + rs = — | ] {r 2 + s 2j rt 2 ) i , 

and E is to be determined from the condition 

(s-r) (r-t) (t-s) = -rihr|p 

Now the square of the product on the left is, by a known theorem, equal to 

- 27 (« 3 + fVP 2 " + E) (/3 3 + T \P 2 /3 + E), 
where «, /3 are the roots of 

d 



dX 



(X3 +1 %P 2 X + E) = 0, 



or, 3X 2 + t VP 2 = 0. 

Hence we obtain 

E 2 = - t JLp 2 3 (40 + I>2 



4 2 27 3 \ Pi 2 , 

The sign of E may be taken +, for the roots 0i, 2 , @3 all lie between 
— a 2 and — c 2 , and if they are arranged in ascending order of numerical 
magnitude, r and t will be positive and s negative. 

(fc v, £). ©1. ©2, ©s. 



1906.] The Calculation of Ellipsoidal Harmonics. 463 

5. The harmonics of seventh degree can be determined in a similar 
manner. As, however, the expressions are somewhat longer to write, it will 
be sufficient to state the leading subsidiary results for one type in such a 
form that similar relations can be easily found for the other two by inter- 
change of letters. The type chosen is £©i© 2 ©3. 

(a) 5 (b 2 + u) (c 2 + u) + 5 (o 2 + u) (a 2 -f u) + 7 {a 2 + %i) (b 2 -hit) = — 14V. 

( B ) 5( s -r)(r-t)(t-s)[4:(a 2 + ti) + 4:(b 2 + ii) + 3(c 2 -i-u)] 

= 4:(a 2 +u)(c 2 +u)Y-7Y 2 . 

(c) 4 (a 2 + u) (b 2 + u) (c 2 + u) + 4Y [5 (a 2 + u) + 5 (b 2 + u) + 4 (a 2 + u)] 

= —5 (s— r){r— t){t— s). 

From these three results the equation for the sum of the roots and, with 
the aid of r-f s + t = 0, the equation for the difference of the roots can be 
readily formed, as in § 4. 

6. To verify results proceed as follows : — Putting p = 0, which is 
•equivalent to making a = 5, equation (1) or (2), for instance, one value of 
6 being thus —a 2 , will give a second between — b 2 and — c 2 , say — -x x (c 2 —a 2 ) 
-and -J^i + ^ ought then to satisfy equation (13) with^? zero. In like manner 
if we put p = 1 or b = c, 6 = — b 2 will be one root and the other will lie 
between —a 2 and ~6 2 , say -x 2 (c 2 —a 2 ) } and — J(l-faj 2 ) + | ought then to 
.satisfy (13) withp unity. By this means the accuracy of the results given 
in § 3 has been tested both when p = and p = 1. 

Further it appears that of the three members of any type of harmonic 
with two © factors one member has both 6 roots between —a 2 and — b 2 , 
another has both between ~b 2 and — c 2 , and the third has one root in one 
compartment and the other in the other. Again, with harmonics with three 
© factors one has all three values of 6 between — a 2 and — b 2 } a second all 
between —5 2 and — c 2 and the other two have respectively one in one 
•compartment and two in the other. 

[Added March 29. Approximations, — The solutions given above are 
applicable whether the ellipsoids are prolate or oblate, but in some of the 
physical problems in which the harmonics under consideration might be 
required, either a is nearly equal to b or b to c, and in those cases the exact 
expressions would be usefully replaced by series in ascending powers of p or 
q(=l—p\p being applicable to a prolate and a to an oblate ellipsoid. 
Taking, for instance, the harmonic © x © 2 , we have already found the equation 
(11), which is suitable to the prolate form. The corresponding equation for 
the oblate is most readily obtained by writing 

c 2 — b 2 f c 2 + u t a 2 +it f b 2 + u = (q, z, z — l f z—q) {c 2 -—a 2 ) 



464 



The Calculation of Ellipsoidal Harmonics. 



in the relation (7). We shall then obtain 

49s 8 - 49(1 + q)z 2 + (12 + 37+12q 2 )z-~6(q + q 2 )=: 0. 

To solve this last in series proceeding in powers of q, observe that the part 
of the equation not involving q may be written z (7z— 4) (7z— -3) and 
proceed by successive approximations. The roots will then be found to be 



k— iW- 






(29) 



> 



(30) 



and the corresponding values of y, for the differences, 

±V / t^(2+---X 

±y / ^(i--k+ffg 2 +AVg 3 +---)> 
± v^i -k- 4f !<? 2 - ^i? 3 + • • •)• 

If we compare the equation in z with that in x, it is clear that if we write 
p instead of q and change the signs of the series for z, we shall get the values 
of x and the corresponding differences. 

This method applies to harmonics of the fifth degree, except that in the 
cases |©iB 2 and f ©i© 2> we shall not have the same simple relations between 
the equations in x and z. 

When p = q = ^ we reach the partition line between prolate and oblate 
ellipsoids, and at this particular point the expressions for the roots are 
simpler.]